TrianglesTriangles

A triangle is a polygon with three sides.

Classifying Triangles

For example,

A

B

C

We name a triangle using its vertices.

∆ABC

∆BAC

∆CAB ∆CBA

∆BCA

∆ACB

Opposite Sides and Angles

A

B

C

We say that is opposite .ABC

What is opposite ?C

What is opposite of ?CA

Triangles can be classified by their

Sides

• Scalene

• Isosceles

• Equilateral

Angles

• Acute

• Right

• Obtuse

• Equiangular

Equilateral TriangleA triangle in which all 3 sides are equal

m AC = 7.00 cm

m BC = 7.00 cmm AB = 7.00 cm

A

B

C

Isosceles TriangleA triangle in which at least 2 sides are equal

m AC = 7.90 cm

m BC = 5.51 cmm AB = 5.51 cm

A

B

C

Scalene TriangleA triangle in which all 3 sides are different lengths

m AC = 8.21 cm

m BC = 6.60 cmm AB = 4.65 cm

A

B

C

Acute Triangle

A triangle in which all 3 angles are less than 90˚

mCBA = 78°

mBCA = 58°

mBAC = 44°

A

B

C

Right TriangleA triangle in which exactly one angle is 90˚

mCBA = 90°

mBCA = 55°

mBAC = 35°

A

B

C

Obtuse TriangleA triangle in which exactly one angle is greater than 90˚and

less than 180˚

mCBA = 119°

mBCA = 40°

mBAC = 21°

A

B

C

Equiangular Triangle

A triangle in which all 3 angles are the same measure.

mCBA = 60°

mBCA = 60°mBAC = 60°

A

B

C

Angles When the sides of a polygon are extended, other anglesare formed. The inside/original angles are the interior angles.The adjacent/outside angles that form linear pairswith the interior angles are the exterior angles.

Interior angles

Exterior angles

3

1

2

5

4

6

<4, <5, <6

<1, <2, <3

TRIANGLE INVESTIGATION

Triangle Sum TheoremThe sum of the interior angles in a triangle is

180˚.

60 80

40

Find the value of x.

Example:

x3x

2x

EXTERIOR TRIANGLE INVESTIGATION

Exterior Angle TheoremThe measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Exterior AngleRemote Interior Angles

A

BC D

m ACD m A m B

Find the value of x.

Example:

x

70

(2x+10)

A corollary to a theorem is a statement thatcan be proven easily using another theorem.

Corollary

Definition:

Third Angle CorollaryIf two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.

Equiangular Corollary

Each angle in an equiangular triangle is 60˚.

Right Angle Corollary

There can be at most one right or obtuse angle in a triangle.

Acute Corollary

Acute angles in a right triangle are complementary.